Quantum Processes (The computational model) Lu´ ıs Soares Barbosa IC May 2019
Qubits | v � = α | u � + β | u ′ � In a sense | u � can be thought as being simultaneously in both states, but be careful: states that are combinations of basis vectors in similar proportions but with different amplitudes, e.g. 1 1 ( | u � + | u ′ � ) ( | u � − | u ′ � ) √ and √ 2 2 are distinct and behave differently in many situations. Amplitudes are not real (e.g. probabilities) that can only increase when added, but complex so that they can cancel each other or lower their probability
The state space of a qubit Representation redundancy: qubit state space � = complex vector space used for representation Global phase Unit vectors equivalent up to multiplication by a complex number of modulus one, i.e. a phase e i θ , represent the same state. Let | v � = α | u � + β | u ′ � | e i θ α | 2 = ( e i θ α )( e i θ α ) = ( e − i θ α )( e i θ α ) = αα = | α | 2 and similarly for β . As the probabilities | α | 2 and | β | 2 are the only measurable quantities, the global phase has no physical meaning.
The state space of a qubit Relative phase Is a measure of the angle between the two complex numbers α and β , cf 1 1 1 ( | u � + | u ′ � ) ( | u � − | u ′ � ) ( e i θ | u � + | u ′ � ) √ √ √ 2 2 2 ... cannot be discarded!
The mathematical framework Complex, inner-product vector space A set U of vectors generates a complex vector space whose elements can be written as linear combinations of vectors in U : | v � = a 1 | u 1 � + a 2 | u 2 � + · · · + a n | u n � i.e. • Abelian group ( V , + , − 1 , 0 ) • with scalar multiplication ( c · | v � distributing over + , often represented by juxtaposition)
The mathematical framework • A inner product � − | − � : V × V − → C such that � � ( 1 ) � v | λ i · | w i �� = λ i � v | w i � i i ( 2 ) � v | w � = � w | v � ( 3 ) � v | v � ≥ 0 (with equality iff | v � = 0) Note: � − | − � is conjugate linear in the first argument: � � � λ i · | w i � | v � = λ i � w i | v � i i Notation: � v | w � ≡ � v , w � ≡ ( | v � , | w � )
The mathematical framework Old friends • | v � and | w � are orthogonal if � v | w � = 0 � • norm: || v � | = � v | v � | v � • normalization: || v � | • | v � is a unit vector if || v � | = 1 • A set of vectors {| i � , | j � , · · · , } is orthonormal if each | i � is a unit vector and � i = j ⇒ 1 � i | j � = δ i , j = otherwise ⇒ 0 Note A basis for V (set of linearly independent elements of V spanning V ) will usually be taken as orthonormal.
The mathematical framework C n The inner product in C n of two vectors over the same orthonormal basis boils down to vector multiplication: � � � v | w � = � v i | i � | w j | j �� i j � = v i w j δ i , j i , j � = v i w i i w 1 � v 1 · · · v n � . . = . w n
The mathematical framework Matrices as linear maps Any m × n matrix M can be seen as a linear operator mapping vectors in C n to vectors in C m . Linearity means that � � = M α j | v j � α j M | v j � j j holds, where the action of M in a m -dimensional vector corresponds to multiplication. Examples: The Pauli matrices � 1 � � 0 � � 0 � � 1 � 0 1 − i 0 I = X = Y = Z = 0 1 1 0 i 0 0 − 1
The mathematical framework Linear maps as matrices Let V and W be vector spaces with basis, respectively, B V = {| v 1 � , · · · , | v n � } and B W = {| w 1 � , · · · , | w m � } A linear operator, i.e. a map M : V − → W st � � = M α j | v j � α j M ( | v j � ) j j can be represented by a m × n matrix st, for each j ∈ 1 .. n , � M ( | v j � ) = M i , j | w i � i Composition of linear operators amounts to multiplication of the corresponding matrices. This representation is, of course, basis dependent.
The mathematical framework Hilbert spaces Complete, complex, inner-product vector space, complete meaning that any Cauchy sequence | v 1 � , | v 2 � , · · · converges ∀ ǫ> 0 ∃ N ∀ m , n > 0 || v m � , | v n � | ≤ ǫ This completeness condition is trivial in finite dimensional vector spaces
Classical systems State spaces in a classical system combine through direct sum: n 2-dimensional vector a vector in 2 n -dimensional vector space � Direct sum V ⊕ W • B V ⊕ W = B V ∪ B W and dim ( V ⊕ W ) = dim ( V ) + dim ( W ) • Vector addition and scalar multiplication are performed in each component and the results added • � ( | u 2 � ⊕ | z 2 � ) | ( | u 1 � ⊕ | z 1 � ) � = � u 2 | u 1 � + � z 2 | z 1 � • V and W embed canonically in V ⊕ W and the images are orthogonal under the standard inner product Example a � a � � c � b ⊕ = b d c d
Quantum systems State spaces in a classical system combine through tensor: a vector in 2 n -dimensional vector space n 2-dimensional vector � i.e. the state space of a quantum system grows exponentially with the number of particles: Feyman’s original motivation Tensor V ⊗ W • B V ⊗ W is a set of elements of the form | v i � ⊗ | w j � , for each | v i � ∈ B V , | w i � ∈ B W and dim ( V ⊗ W ) = dim ( V ) × dim ( W ) • ( | u 1 � + | u 2 � ) ⊗ | z � = | u 1 � ⊗ | z � + | u 2 � ⊗ | z � • | z � ⊗ ( | u 1 � + | u 2 � ) = | z � ⊗ | u 1 � + | z � ⊗ | u 2 � • ( α | u � ) ⊗ | z � = | u � ⊗ ( α | z � ) = α ( | u � ⊗ | z � ) • � ( | u 2 � ⊗ | z 2 � ) | ( | u 1 � ⊗ | z 1 � ) � = � u 2 | u 1 �� z 2 | z 1 �
Assembling through ⊗ Clearly, every element of V ⊗ W can be written as α 1 ( | v 1 � ⊗ | w 1 � ) + α 2 ( | v 2 � ⊗ | w 1 � ) + · · · + α nm ( | v n � ⊗ | w m � ) Example The basis of V ⊗ W , for V , W qubits with the standard basis is {| 0 � ⊗ | 1 � , | 0 � ⊗ | 1 � , | 1 � ⊗ | 0 � , | 1 � ⊗ | 1 � } Thus, the tensor of α 1 | 0 � + β 1 | 1 � and α 2 | 0 � + β 2 | 1 � α 1 α 2 | 0 � ⊗ | 0 � + α 1 β 2 | 0 � ⊗ | 1 � + α 2 β 1 | 1 � ⊗ | 0 � + α 2 β 2 | 1 � ⊗ | 1 � In a simplified notation α 1 α 2 | 00 � + α 1 β 2 | 01 � + α 2 β 1 | 10 � + α 2 β 2 | 11 �
Entanglement Most states in V ⊗ W cannot be written as | u � ⊗ | z � • A single-qubit state can be specified by a single complex number so any tensor product of n qubit states can be specified by n complex numbers. But it takes 2 n − 1 complex numbers to describe states of an n qubit system. • Since 2 n ≫ n , the vast majority of n -qubit states cannot be described in terms of the state of n separate qubits. • Such states, that cannot be written as the tensor product of n single-qubit states, are entangled states.
Entanglement Example The Bell state | Φ + � = 1 2 ( | 00 � + | 11 � ) is entangled √ Actually, to make | Φ + � equal to ( α 1 | 0 � + β 1 | 1 � ) ⊗ ( α 2 | 0 � + β 2 | 1 � ) = α 1 α 2 | 00 � + α 1 β 2 | 01 � + β 1 α 2 | 10 � + β 1 β 2 | 11 � would require that α 1 β 2 = β 1 α 2 = 0 which implies that either α 1 α 2 = 0 or β 1 β 2 = 0. Note Entanglement can also be observed in simpler structures, e.g. relations: { ( a , a ) , ( b , b ) } ⊆ A × A cannot be separated, i.e. written as a Cartesian product of subsets of A .
Entanglement The notion of entanglement • is not basis dependent • but depends on the tensor decomposition used Example. u = 1 2 ( | 0000 � + | 0101 � + | 1010 � + | 1111 � ) is entangled wrt the decomposition into single qubits, since it cannot be expressed as the tensor product of four single-qubit states, but it is not for a decomposition consisting of a subsystem of the first and third qubit and another with the second and fourth qubit: 1 1 u = √ ( | 0 1 0 3 � + | 1 1 1 3 � ) ⊗ √ ( | 0 2 0 4 � + | 1 2 1 4 � ) 2 2
Dirac’s notation Dirac’s bra/ket notation is a handy way to represent elements and constructions on an Hilbert space, amenable to calculations and with direct correspondence to diagrammatic (categorial) representations of process theories | u � A ket stands for a vector in an Hilbert space V . In C n , a column vector of complex entries. The identity for + (the zero vector) is just written 0. � u | A bra is a vector in the dual space V † , i.e. scalar-valued linear maps in V — a row vector in C n . There is a bijective correspondence between | u � and � u | u 1 � u 1 · · · u n � . | u � = . = � u | ⇔ . u n A tradition going back to Penrose in the 1970’s.
Dirac’s notation Dirac’s bra/ket notation provides a convenient way of specifying linear transformations on quantum states: outer product | w �� u | ( | z � ) � = | w �� u || z � = | w � � u | z � = � u | z � | w � • matrix multiplication (composition of linear maps) is associative and scalars (zero objects in the corresponding universe) commute with everything
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